CN116490828A - Method and configuration system for configuring a control device for a technical system - Google Patents

Method and configuration system for configuring a control device for a technical system Download PDF

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CN116490828A
CN116490828A CN202180071824.6A CN202180071824A CN116490828A CN 116490828 A CN116490828 A CN 116490828A CN 202180071824 A CN202180071824 A CN 202180071824A CN 116490828 A CN116490828 A CN 116490828A
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configuration data
data set
performance
ctl
pareto
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S·费勒
D·海恩
M·托基奇
S·乌德鲁夫特
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Siemens AG
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    • GPHYSICS
    • G05CONTROLLING; REGULATING
    • G05BCONTROL OR REGULATING SYSTEMS IN GENERAL; FUNCTIONAL ELEMENTS OF SUCH SYSTEMS; MONITORING OR TESTING ARRANGEMENTS FOR SUCH SYSTEMS OR ELEMENTS
    • G05B13/00Adaptive control systems, i.e. systems automatically adjusting themselves to have a performance which is optimum according to some preassigned criterion
    • G05B13/02Adaptive control systems, i.e. systems automatically adjusting themselves to have a performance which is optimum according to some preassigned criterion electric
    • G05B13/0265Adaptive control systems, i.e. systems automatically adjusting themselves to have a performance which is optimum according to some preassigned criterion electric the criterion being a learning criterion
    • GPHYSICS
    • G06COMPUTING; CALCULATING OR COUNTING
    • G06FELECTRIC DIGITAL DATA PROCESSING
    • G06F30/00Computer-aided design [CAD]
    • G06F30/20Design optimisation, verification or simulation
    • G06F30/27Design optimisation, verification or simulation using machine learning, e.g. artificial intelligence, neural networks, support vector machines [SVM] or training a model
    • GPHYSICS
    • G06COMPUTING; CALCULATING OR COUNTING
    • G06FELECTRIC DIGITAL DATA PROCESSING
    • G06F30/00Computer-aided design [CAD]
    • G06F30/20Design optimisation, verification or simulation
    • G06F30/25Design optimisation, verification or simulation using particle-based methods
    • GPHYSICS
    • G06COMPUTING; CALCULATING OR COUNTING
    • G06FELECTRIC DIGITAL DATA PROCESSING
    • G06F2111/00Details relating to CAD techniques
    • G06F2111/06Multi-objective optimisation, e.g. Pareto optimisation using simulated annealing [SA], ant colony algorithms or genetic algorithms [GA]

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Abstract

In order to configure the control device (CTL), a predefined standard configuration data set (L0) is read in. Furthermore, deviations from the standard configuration data set (L0) and control performances are determined for the plurality of generated test configuration data sets (LT). Furthermore, pareto optimization is performed for a large number of test configuration data sets (LT), wherein the deviation and the control performance are used as pareto target criteria. A configuration data set (LPO) resulting from pareto optimization is then selected for configuring the control means (CTL).

Description

Method and configuration system for configuring a control device for a technical system
Background
Complex technical systems, such as traffic signal systems, turbines, production plants, robots or motors, often require complex configurations for productive operation in order to optimize the performance of the technical system in a targeted manner. The performance to be optimized may for example relate to capacity, yield, resource requirements, efficiency, harmful substance emissions, stability, wear and/or other target parameters of the technical system.
Modern control devices of technical systems often use data-driven machine learning methods for optimized configuration. By means of such a learning method, the control device can be trained to determine those control actions which specifically lead to a desired or otherwise optimal behavior of the technical system on the basis of the current operating data of the technical system. For this purpose, a large number of known machine learning methods, in particular reinforcement learning methods, are available.
However, in many cases, the configuration found by the performance driven optimization method should still be verified or validated in terms of its security and/or its user acceptance.
Control optimizations for technical systems are known from publication WO2016/000851A1, in which the effect of configuration interventions can be interactively determined by means of simulation. However, such interactive verification often incurs relatively high manual costs.
Disclosure of Invention
The object of the present invention is to specify a method and a configuration system for configuring a control device for a technical system, which allow for an improved safety and/or user acceptance of the configuration with less effort.
This object is achieved by a method having the features of patent claim 1, by a configuration system having the features of patent claim 13, by a computer program product having the features of patent claim 14 and by a computer readable storage medium having the features of patent claim 15.
For configuring a control device for a technical system, a predefined standard configuration data set for the control device is read in. In this case, the technical system may be, in particular, a traffic signal system, a turbine, a production plant, a robot, an electric motor, another machine, another device or another apparatus. In addition, a plurality of measurement configuration data sets are generated. According to the invention, for the respective test configuration data set, a deviation value is determined which quantifies a deviation from the standard configuration data set and a performance value is quantified for controlling a performance of the technical system in accordance with the respective test configuration data set. Furthermore, pareto optimization is performed for a large number of test configuration data sets, wherein the bias and performance are used as pareto target criteria. As is well known, pareto optimization is a multi-criterion optimization with a plurality of target criteria, also referred to herein and below as pareto target criteria. The configuration data set resulting from the pareto optimization is then selected for configuring the control device.
A configuration system, a computer program product and a computer-readable, preferably non-volatile, storage medium are provided for carrying out the method according to the invention.
The method according to the invention and the configuration system according to the invention may be performed or implemented, for example, by means of one or more computers, processors, application Specific Integrated Circuits (ASICs), digital Signal Processors (DSPs) and/or so-called "field programmable gate arrays" (FPGAs).
As long as the deviation from the standard configuration data set and thus the similarity to the standard control behavior are considered to some extent in pareto optimization, a configuration that is generally both good performing and has a high user acceptance can be determined. Furthermore, non-optimal configurations can be excluded in a simple manner by pareto optimization.
Advantageous embodiments and developments of the invention are specified in the dependent claims.
According to an advantageous embodiment of the invention, the data elements of the standard configuration data set may be selected. A test configuration data set may then be generated from the standard configuration data set, wherein variations in the selected data elements are suppressed. In particular, selected data elements may be taken over unchanged into the corresponding standard configuration data set, while other data elements of the standard configuration data set are shifted. Thus, an impermissible or undesired change of configuration can be excluded in a simple manner, or compliance with boundary conditions can be ensured.
The pareto front may advantageously be determined by pareto optimization within the generated test configuration data set. The configuration data set may then be selected from the pareto front for configuring the control device. A collection of configuration data sets, which are, for example, not more than a given threshold value from a mathematically exact pareto optimum value, is also to be understood as a pareto front. By limiting the selection to the pareto front, the space of possible configurations is often significantly limited, wherein in particular non-optimal configurations are cancelled. The subsequent optimization of the configuration data set of the selection and, if necessary, of the pareto front is thus significantly simplified.
In particular, the pareto optimization may be performed using genetic optimization methods, genetic programming methods, gradient-based optimization methods, random gradient methods, particle swarm optimization, metapolis optimization methods, and/or other machine learning methods. A number of highly efficient standard routines are available for the mentioned optimization method.
Advantageously, a new set of configuration data generated when performing pareto optimization may be used as the test set of configuration data. In this case, new configuration data sets can be generated in the context of performance-driven optimization. In this way, the generation of the test configuration data set may be driven, preferably in the direction of the high performance configuration.
According to an advantageous embodiment of the invention, for determining the performance values for the respective test configuration data sets, the technical system and/or a simulation model of the technical system can be controlled in dependence on the respective test configuration data sets, wherein the resulting performance of the technical system is measured. A proxy model of the technical system may preferably be used as a simulation model, which requires less computational resources than a complete simulation.
According to a preferred embodiment of the invention, in order to determine the performance values for the respective test configuration data sets, a deviation of the response behaviour of the control device configured with the respective test configuration data sets from the response behaviour of the control device configured with the performance-optimized configuration data sets can be determined. The performance-optimized configuration data set can be used to some extent as a benchmark for achievable performance and can preferably be determined by means of reinforcement learning methods. Such reinforcement learning methods are often also referred to as reinforcement learning (Reinforcement Learning).
According to a further embodiment of the invention, in order to determine the deviation value for the respective test configuration data set, a deviation of the component representation of the respective test configuration data set from the component representation of the standard configuration data set may be determined. The component representation may be a vector representation, for example. In the case of genetically encoded configuration data sets, the bias may be expressed by the number of differences in the genome of the relevant configuration data set. I.e. to some extent by deviations in the genotypes of the configuration data set. In particular, the deviation may be expressed by a minimum number of operations required to translate one of the configuration data sets into another configuration data set.
Furthermore, in order to determine deviation values for the respective test configuration data sets, deviations of the response behaviour of the control devices configured with the respective test configuration data sets from the response behaviour of the control devices configured with the standard configuration data sets may be determined. Such deviations may be understood to some extent as deviations in the phenotype of the configuration data set.
Drawings
Embodiments of the present invention are described in more detail below with reference to the accompanying drawings. In each case shown in schematic form:
fig. 1 shows a traffic signal system with a control device, and
fig. 2 shows a configuration system according to the invention when configuring a control device.
Detailed Description
Fig. 1 shows in a schematic illustration a traffic signal system arranged at an intersection as a technical system TS, which is coupled to a device controller CTL as a control device for the traffic signal system TS. The traffic signal system TS has a signal set SG, which may comprise one or more traffic signals, and a sensor system S for continuously measuring and/or detecting operating parameters of the traffic signal system TS and local traffic data.
Alternatively, the technical system TS may also comprise turbines, production equipment, robots, motors, 3D printers, other machines, other devices or other equipment. In such a case, the method for configuring the control device CTL according to the invention can be applied in a similar manner.
The device controller CTL is computer-assisted configurable and may be implemented as part of the traffic signal system TS or may be implemented wholly or partly outside the traffic signal system TS. The device controller CTL is used to control the traffic signal system TS in a traffic-related manner.
The device controller CTL should be configured such that the technical system TS is controlled in an optimized manner. The control of the technical system (here TS) is also to be understood in this case as meaning the regulation of the technical system and the output and use of control-related, i.e. data and control signals which contribute to the targeted influence of the technical system.
In the present embodiment, the device controller CTL should be configured by means of a machine learning method such that the control of the signal phase (Signalphasen) of the signal group SG or the traffic light (lichtzeichenange) TS or other traffic control actions is optimized in accordance with the detected operating parameters and traffic data. Here and in the following, the term optimization should also be understood as being close to the optimal value. In particular, the waiting time of the vehicle should be reduced and/or the throughput of the vehicle should be increased. Alternatively or additionally, other parameters contributing to the performance of the traffic light TS or the device controller CTL may also be optimized.
The control behavior of the device controller CTL is specified by its configuration. In order to configure the device controller CTL in an optimized manner, an optimized configuration data set LPO is transmitted to the device controller CTL, by means of which optimized configuration data set LPO a number of setting parameters of the device controller CTL are specifically set. The control behavior thus set by the device controller CTL is often also referred to as policy or response behavior. The optimized configuration data set LPO is determined by the method according to the invention.
Such a configuration data set specifying the control behavior of the device controller CTL may be represented by a different data structure. Accordingly, the corresponding configuration data set may include a number of parameters, variables or action selection rules, program code, syntax trees, mathematical expressions, classifiers, neural weights, PID modulators (PID: proportional Integral Derivative (proportional-integral-derivative)), other modulators, and/or other descriptive data for configuration.
Continuously detected traffic data and operating parameters of the traffic light TS are transmitted by the sensor system S to the device controller CTL in the form of sensor data SD. The operating parameters may include operating states for the traffic light TS, for example, for the current traffic light phase, for the switching state, for the control action, for the system state and/or for the system characteristic. The traffic data may include, in particular, instructions on the number and/or speed of the vehicle, on the current traffic load and/or on the local harmful substance load.
From the transmitted sensor data SD, control data CD is generated by the device controller CTL, which are transmitted from the device controller CTL to the traffic light TS for controlling said traffic light TS in an optimized manner. The control data CD is generated according to the policy of the device controller CTL configured by the optimized configuration data set LPO.
Fig. 2 shows a schematic illustration of a configuration system KS according to the invention when a control device CTL is configured. Wherever the same or corresponding reference numerals are used in fig. 2 as in fig. 1, these reference numerals denote the same or corresponding entities, which may in particular be realized or designed as described above.
The control means CTL may form part of the configuration system KS or may be arranged wholly or partly outside the configuration system KS. The configuration system KS and/or the control means CTL have one or more processors for implementing the method according to the invention and one or more memories for storing data to be processed. As already mentioned above, the control device CTL should be configured as a technical system TS, for example a traffic signal system, by means of the configuration system KS in an optimized manner.
As a starting point for the configuration, there is a standard configuration data set L0, which is entered by the user USR of the configuration system KS or read in from a database. The standard configuration data set L0 specifies a standard configuration of the control device CTL, by means of which the control device CTL reacts to the sensor system and controls the technical system TS accordingly. Standard control actions resulting from standard configurations are generally usable and validated, but are not yet optimal.
Furthermore, the specification EL of the data element of the standard configuration data set L0 to be kept constant during the optimal configuration is read in by the user USR or from the database. Thus, the remaining data elements of the standard configuration data set L0 may be changed within the scope of the optimization. In this way, impermissible or untrusted changes in control behavior with respect to standard configurations may be excluded. In the case of traffic lights as technical system TS, the duration of the yellow light phase or the minimum duration for the green light phase or the red light phase can thus be kept constant.
The configuration achievable by varying the data elements of the standard configuration data set L0 that are not to be kept constant constitutes a configuration space that can be used optimally.
The standard configuration data set L0 and the description EL are transmitted to the deviation evaluator EVD of the configuration system KS for initializing the deviation evaluator EVD.
The deviation evaluator EVD serves to quantify the deviation of the response behavior specified thereby or thereby from the standard configuration for the respective configuration data set. The corresponding similarity to the standard configuration can be expressed, in particular, by a corresponding deviation. Such similarity may be understood herein to some extent as the closeness of a given configuration or resulting control behavior. The high affinity (Vertrautheit) of controlling behavior also generally improves its user acceptance and/or its security. In the case of a traffic signal system as technical system TS, a configuration similar to the standard configuration is more likely to meet the expectations of traffic participants than a less similar configuration.
Furthermore, a performance-optimized configuration data set LV for the control means CTL is determined by the first optimizer OPTRL. For this purpose, the first optimizer OPTRL has a simulation model SIM that models and simulates the technical system TS in terms of its control. In the case of traffic lights as technical system TS, the simulation model SIM simulates the relevant intersections and traffic flows. The performance-optimized configuration data set LV is preferably determined from the simulation model SIM by means of a reinforcement learning method.
The performance in view of the optimization may relate to, inter alia, the capacity (Leistung) of the technical system TS, profits, speed, time requirements, running time, resource requirements, efficiency, accuracy, stability, wear, service life, harmful substance emissions, traffic throughput and/or error rate. A number of known and efficient reinforcement learning methods are available for implementing such performance driven optimizations.
The optimization is performed by the first optimizer OPTRL without taking into account deviations from the standard configuration data set L0, so that the performance-optimized configuration data set LV may lead to completely unreliable control behavior and may lie outside the allowed configuration space. Thus, the performance-optimized configuration data set LV cannot generally be used directly for configuring the control device CTL, but rather forms a comparison benchmark (vergleichsman beta stab) for the fully achievable performance of the control device CTL or of the technical system TS within the scope of the invention.
The data set DS is generated by means of a reinforcement learning method performed by the first optimizer OPTRL, said data set comprising the realized system state of the technical system TS, the performed control actions, the subsequent states realized thereby and the resulting rewards for the success of the quantized control actions in the sense of reinforcement learning.
The performance optimized configuration data set LV and the data set DS are transferred from the first optimizer OPTRL to the performance evaluator EVP of the configuration system KS.
The performance evaluator EVP serves to quantify the performance of the technical system TS controlled in accordance with the configuration data set for the respective configuration data set. In this way, the control performance of the control device CTL thus configured is evaluated to some extent. Here, the performance-optimized configuration data set LV is used as a comparison criterion for the achievable performance by the performance evaluator EVP.
The configuration system KS furthermore has a second optimizer which is designed as a Pareto optimizer OPTP for carrying out Pareto-optimization (Pareto-optimizer). Pareto optimization is a multi-criterion optimization in which a plurality of different target criteria, the so-called pareto target criteria, are considered independently. As a result of Pareto optimization, a so-called Pareto Front (Pareto-Front) PF is determined. Such pareto fronts are often also referred to as pareto sets. The pareto front (here PF) is a solution to the multi-criterion optimization problem in that one of the target criteria cannot be improved without degrading the other target criteria. Thus, the pareto front forms to some extent the best compromise set. In particular, solutions not included in the pareto front PF can still be improved in view of at least one target criterion. Thus, by limiting to the pareto front PF, eliminating a large number of solutions is certainly not optimal. As long as the pareto front usually only comprises a very small part of the possible solution space, by limiting to the pareto front, the subsequent selection or further optimization costs are significantly reduced.
According to the invention, the pareto target criterion regarding deviation from the standard configuration and performance is determined by the pareto optimizer OPTP as the pareto front PF. In this case, the optimization is carried out in the direction of greater performance and smaller deviations, i.e. greater intimacy of the control behavior. A number of standard routines, especially machine learning methods, are available for such pareto optimization.
In this example, genetic programming (genetische Programmierung) methods are used for pareto optimization. In the scope of genetic programming, an optimized configuration is sought in the allowed configuration space. For this purpose, the generator GEN of the pareto optimizer OPTP generates a large number of new configuration data sets which are used as test configuration data sets LT. To initialize the generator GEN, the standard configuration data set L0 and the specification EL of the data elements to be kept constant for the standard configuration data set L0 are transmitted to the generator GEN. The test configuration data set LT is generated by the generator GEN from the standard configuration data set L0, wherein the data elements of the standard configuration data set L0 identified by the specification EL are not changed, while the remaining data elements thereof are changed in the permitted configuration space.
The generated test configuration data set LT is transmitted from the generator GEN to the performance evaluator EVP and to the deviation evaluator EVD. As already indicated above, the performance evaluator EVP determines for the respective test configuration data set LT a respective performance value PW which quantifies the performance of the technical system TS controlled in accordance with this test configuration data set LT. For this purpose, in the present embodiment, a deviation of the response behavior of the control device CTL configured with the corresponding test configuration data set LT from the response behavior of the control device CTL configured with the performance-optimized configuration data set LV is determined. Here, the system states contained in the data set DS are used as representative control points for which the relevant response behavior is compared. Here, the closer the response behavior of the test configuration data set LT is to the response behavior of the performance-optimized configuration data set LV, the higher the calculated performance value PW.
Alternatively or additionally, the performance of the respective test configuration data set LT can also be determined in that the control behavior thus specified is simulated for a large number of time steps by means of a simulation model of the technical system TS or of the control device CTL thereof. Here, the jackpot or cumulative benefit of the simulated behavior is measured. In this case, a so-called proxy model (Surrogat-model) of the technical system TS is preferably used as simulation model, which requires less computational resources than detailed simulation.
The respective performance values PW determined for the respective test configuration data set LT are transferred from the performance evaluator EVP to the pareto optimizer OPTP. The transmitted performance values PW are used by the pareto optimizer OPTP in the sense of genetic programming as fitness of the corresponding test configuration data set LT. Thus, the performance evaluator EVP implements a fitness function (fitnessfung) that evaluates the performance for the test configuration data set LT.
Genetic programming produces a large number of test configuration data sets LT of varying fitness. Genetic generation is preferably driven in the direction of a good performing configuration data set by an fitness function evaluating performance. However, multi-criterion optimization is implemented, wherein the deviation from the standard configuration data set L0 is used as an independent optimization metric.
The last-mentioned deviation is determined by a deviation evaluator EVD. As already indicated above, the deviation evaluator EVD determines for the respective test configuration data set LT a respective deviation value D which quantifies the deviation between the respective test configuration data set LT and the standard configuration data set L0 and thus the similarity thereof.
The deviation value may be determined as, for example, a euclidean distance in a vector space represented by components of the configuration data sets LT and L0 as d= |lt-l0| or d= (LT-L0) 2 . In the case of genetically encoded configuration data sets LT and L0, the deviation value D may be expressed by the number of differences in the genomes of the configuration data sets LT and L0. In particular by the minimum number of operations necessary to translate the standard configuration data set L0 into the test configuration data set LT. If the configuration data sets L0 and LT are encoded as syntax trees, the deviation value D may be determined as a so-called Tree Edit Distance (Tree-Edit-Distance).
Alternatively or additionally, the corresponding response behavior of the test configuration data set LT can also be compared with the response behavior of the standard configuration data set L0, and the deviation value D can be calculated therefrom.
The deviation value D is transmitted from the deviation evaluator EVD to the pareto optimizer OPTP. The pareto front PF is determined within the generated test configuration data set LT by the pareto optimizer OPTP from the received deviation value D and the performance value PW.
Finally, an optimized configuration data set LPO is selected from the resulting pareto front PF. If necessary, predefined selection criteria, in particular one or more further optimization criteria, can still be applied when selecting the optimized configuration data set LPO. By limiting the selection or subsequent optimization to the pareto front PF, the space of possible configurations is often significantly limited, wherein in particular non-optimal configurations are cancelled. The selection of the optimized configuration data set LPO or even other optimizations is thus significantly simplified.
The optimized configuration data set LPO is output as specified for configuring the control device CTL and/or is directly transmitted to the control device CTL in order to configure the control device for controlling the technical system TS in an optimized manner. The resulting configuration of the control means CTL results in a control behaviour which has both high performance and high user acceptance and/or safety. Furthermore, boundary conditions to be observed can be defined in a simple manner by means of the description EL.

Claims (15)

1. A computer-implemented method for configuring a control device (CTL) for a Technical System (TS), wherein
a) Reading in a predefined standard configuration data set (L0) for the control device (CTL),
b) A plurality of test configuration data sets (LT) are generated,
c) Configuration data set (LT) for the corresponding test:
-determining a deviation value (D) quantifying a deviation from said standard configuration data set (L0), and
determining a performance value (PW) quantifying a performance for controlling the technical system according to the respective test configuration data set (LT),
d) Performing a pareto optimization on the plurality of measurement configuration data sets (LT), wherein the bias and the performance are used as pareto target criteria, an
e) -selecting a set of configuration data (LPO) resulting from said pareto optimization for configuring said control means (CTL).
2. The method according to claim 1, characterized in that the data elements of the standard configuration data set (L0) are selected, and
-generating said test configuration data set (LT) from said standard configuration data set (L0), wherein variations of selected data elements are suppressed.
3. The method according to any of the preceding claims, characterized in that the Pareto Front (PF) is determined by pareto optimization within the generated test configuration data set (LT), and
-selecting a configuration data set (LPO) from said Pareto Front (PF) for configuring said control means (CTL).
4. The method according to any of the preceding claims, characterized in that,
the pareto optimization is performed by means of genetic optimization methods, genetic programming methods, gradient-based optimization methods, random gradient methods, particle swarm optimization, metapolis optimization methods, and/or other machine learning methods.
5. Method according to any of the preceding claims, characterized in that a new configuration data set generated when performing the pareto optimization is used as test configuration data set (LT).
6. The method of claim 5, wherein the new configuration data set is generated in a range of performance driven optimizations.
7. Method according to any one of the preceding claims, characterized in that, for determining a performance value (PW) for a respective test configuration data set (LT), the Technical System (TS) and/or a simulation model of the Technical System (TS) is controlled in accordance with the respective test configuration data set (LT), and the resulting performance of the Technical System (TS) is measured there.
8. Method according to any of the preceding claims, characterized in that for determining the performance value (PW) for a respective test configuration data set (LT), a deviation of the response behaviour of the control device (CTL) configured with the respective test configuration data set (LT) from the response behaviour of the control device (CTL) configured with the performance-optimized configuration data set (LV) is determined.
9. Method according to claim 5, characterized in that the performance-optimized configuration data set (LV) is determined by means of a reinforcement learning method.
10. Method according to any of the preceding claims, characterized in that for determining a deviation value (D) for a respective test configuration data set (LT), a deviation of a component representation of the respective test configuration data set (LT) from a component representation of the standard configuration data set (L0) is determined.
11. Method according to any of the preceding claims, characterized in that for determining the deviation value (D) for a respective test configuration data set (LT), a deviation of the response behaviour of the control means (CTL) configured with the respective test configuration data set (LT) from the response behaviour of the control means (CTL) configured with the standard configuration data set (L0) is determined.
12. The method according to any of the preceding claims, characterized in that the Technical System (TS) is a traffic signal system, a turbine, a production plant, a robot, a motor, other machines, other devices or other equipment.
13. A configuration system (KS) for configuring a control device (CTL) for a Technical System (TS), the configuration system being set up for carrying out the method according to any one of the preceding claims.
14. A computer program product arranged to perform the method according to any of claims 1 to 12.
15. A computer readable storage medium having the computer program product of claim 14.
CN202180071824.6A 2020-10-20 2021-09-10 Method and configuration system for configuring a control device for a technical system Pending CN116490828A (en)

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JP5282493B2 (en) * 2008-09-04 2013-09-04 富士通株式会社 Optimal solution relation display device, method, and program
US7996344B1 (en) * 2010-03-08 2011-08-09 Livermore Software Technology Corporation Multi-objective evolutionary algorithm based engineering design optimization
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